Tell what property allows you to compute [tex]$\frac{1}{3} \times\left(6 \times \frac{4}{3}\right)$[/tex] as [tex]$\left(\frac{1}{3} \times 6\right) \times \frac{4}{3}$[/tex]
Answer (1)
The associative property of multiplication allows changing the grouping of factors without changing the result.
For any real numbers a, b, and c: ( a × b ) × c = a × ( b × c ) .
In the given expression, the associative property allows us to compute the product in either order.
The property that allows the given computation is the a ssoc ia t i v e p ro p er t y o f m u lt i pl i c a t i o n .
Explanation
Understanding the Problem The question asks us to identify the property that allows us to change the grouping of factors in a multiplication problem without changing the result.
Stating the Associative Property The property that allows us to re-group factors in multiplication without changing the result is the associative property of multiplication. The associative property states that for any real numbers a, b, and c, the following equation is true: ( a × b ) × c = a × ( b × c ) In simpler terms, it doesn't matter which pair of numbers you multiply first; the final product will be the same.
Applying the Associative Property to the Expression In our case, we have: 3 1 × ( 6 × 3 4 ) = ( 3 1 × 6 ) × 3 4 Here, a = 3 1 , b = 6 , and c = 3 4 . The associative property allows us to compute the expression in either way and get the same result.
Conclusion Therefore, the property that allows us to compute 3 1 × ( 6 × 3 4 ) as ( 3 1 × 6 ) × 3 4 is the associative property of multiplication.
Examples
Imagine you are calculating the total cost of buying multiple items at a store. If you buy 3 items, each costing $2, and you have a 10% discount on the total, you can calculate it in two ways: first find the total cost before the discount (3 * $2) and then apply the discount, or first apply the discount to the cost of one item ($2 * 0.9) and then multiply by the number of items. The associative property ensures that both methods give you the same final cost. This principle applies in various scenarios, from calculating volumes in geometry to determining probabilities in statistics, where the order of operations can be rearranged for convenience without affecting the outcome.
